Investigating Student Understanding of Physics Concepts and the Underlying Calculus Concepts in Thermodynamics
Investigating Student Understanding of Physics Concepts and the Underlying Calculus Concepts in Thermodynamics
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调查学生对物理概念和热力学中基本微积分概念的理解
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发表时间：
2010
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通讯作者：
D. B. Mountcastle
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作者：
John R. Thompson;W. Christensen;D. B. Mountcastle
作为学生理解高级热物理概念的一部分，我们探索学生对物理有效推理所需数学的理解。通过分析学生在回答有关热力学功的概念物理问题以及剥离了物理意义的类似数学问题时使用数学的情况，我们发现有证据表明，学生进入高级物理课程时经常缺乏假定的必备数学知识和/或在物理背景下有效应用数学知识的能力。这些结果表明高级学生没有将微积分和物理纳入一个连贯的框架。我们已经扩展了我们的工作范围，包括在多变量微积分课程结束时对数学概念进行评估。结果支持了物理系学生的发现，即一些观察到的数学困难不仅仅是将数学知识转移到物理环境中，而且似乎源于对数学概念本身的理解。简介 当研究人员接受物理学习和教学调查培训时，作者发现自己面临着学生如何思考微积分概念的问题，这些概念对于对高级热力学课程内容的功能性理解至关重要。我们感兴趣的具体问题的答案在这两个学科研究领域的文献中都没有，这引发了我们的写作问题，并导致我们参加了这次会议。学生对物理和微积分概念的理解 2 热力学主题涵盖了广泛的自然基础领域，并应用于物理、工程和其他自然科学的许多分支。大学层面发表的研究虽然很少，但清楚地表明学生在学习热物理主题时表现出重大困难，包括热、温度、理想气体定律和热力学第一定律的概念困难（Rozier 和 Viennot，1991；Yeo 和 Zadnik，2001；Jasiem 和 Oberem，2002；Loverude 等，2002；Meltzer，2004；Kautz 等， 2005a；考茨等人，2005b)。热力学第一定律将能量转移（即进入或离开相关系统的能量）与某些热力学过程（例如，保持固定温度的气缸中气体的压缩）期间该系统内能的变化联系起来。系统的内能是该系统平衡状态的一个属性，即它取决于系统的温度、压力和体积等量。内能和其他具有类似特征的量被称为状态函数。另一方面，第一定律中的能量传递量、功（机械传递的能量）和热量（热传递的能量）不是状态函数；它们仅在系统正在进行某种过程时才存在。这里重要的一点是，状态函数的变化与热力学过程无关——例如，只需知道初始状态和最终状态即可确定系统内能的变化。相反，系统完成的工作量（或由系统完成的工作量）确实取决于流程。先前关于学生对第一定律的理解的结果记录了一些常见的概念困难，例如将状态函数的概念不加区别地应用于非状态函数量，例如功和热量（Loverude et al., 2002; Meltzer 2004）。学生对物理和微积分概念的理解 3 多年来，缅因大学 (UMaine) 物理教育研究实验室的成员一直在探索学生在主要由物理专业学生学习的高级热物理课程中的学习，将入门物理学的研究工具和结果扩展到更专业的高级部门。该领域的研究有限（Loverude 等人，2002；Meltzer，2004；Meltzer，2005b；Thompson 等人，2006；Cochran 和 Heron，2006；Bucy 等人，2007；Mountcastle 等人，2007；Pollock 等人，2007；Christensen 等人， 2009；洛夫鲁德 2009；史密斯等人，2009）。 Meltzer (2004) 指出，在入门阶段遇到的特殊困难在本科阶段的高级阶段也很明显。在热物理中，与大多数物理领域一样，需要特定的数学概念才能完全理解和理解物理。我们热物理学习和教学的主要研究议程涉及识别和解决学生对物理内容的概念性困难。这里强调的这项研究的一个子主题是调查任何数学概念困难可能在多大程度上影响学生对热力学中相关物理概念的理解。在这个领域，与许多物理领域一样，需要特定的数学概念才能完全理解和理解物理。在这种情况下，我们的研究问题是：学生在多大程度上认识和/或理解物理概念与基础数学之间的关系？数学是解决许多物理问题的重要组成部分，通常用于将复杂的概念问题浓缩为相对简单的变量之间的关系。物理学中存在许多方便的表征工具（例如方程、图表和图表），可以简化复杂问题的分析。对这些表示的适当解释《学生对物理和微积分概念的理解 4》需要认识到相关数学概念的表示和后续应用中内置的物理和数学之间的联系（Redish，2005）。 Meltzer (2002) 展示了数学敏锐度与代数物理课程成功之间的联系。 PER 中只有少数研究试图调查物理学生中微积分概念的数学困难（Thompson 等人，2006；Black 和 Wittmann，2007；Bucy 等人，2007；Pollock 等人，2007；Rebello 等人，2007。）在热力学中，物理过程的二维或三维图形表示对于帮助理解这些过程特别有用。这些图表可以包含有关过程中遵循的热力学“路径”、不同相的区域和关键行为的信息。压力与体积的关系图（称为 P-V 图）被广泛用作物理过程以及相应数学模型的表示。可以从这些表示中快速、轻松地获取信息。例如，在经历热力学过程的系统上所做的功被定义为压力相对于体积的积分：
As part of work on student understanding of concepts in advanced thermal physics, we explore student understanding of the mathematics required for productive reasoning about the physics. By analysis of student use of mathematics in responses to conceptual physics questions about thermodynamic work, as well as analogous math questions stripped of physical meaning, we find evidence that students often enter upper-level physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. These results suggest advanced students are not incorporating calculus and physics into a coherent framework. We have extended our work to include assessment of mathematical concepts at the end of a multivariable calculus course. Results support the findings among physics students that some observed mathematical difficulties are not just with transfer of math knowledge to physics contexts, but seem to have origins in the understanding of the math concepts themselves. Introduction As researchers trained to investigate the learning and teaching of physics, the authors found themselves faced with questions about how students think about calculus concepts that were essential for a functional understanding of the content in an upper-level thermodynamics course. The answers to the specific questions we were interested in were not available in the literature of either discipline-based research field, which prompted our writing questions and has led to our participation in this conference. Student Understanding of Physics and Calculus Concepts 2 The topic of thermodynamics covers a wide-ranging area that is fundamental in nature, and is utilized in many branches of physics, engineering and other natural sciences. The published research at the university level, albeit scarce, clearly indicates that students exhibit significant difficulties when learning thermal physics topics, including conceptual difficulties with heat, temperature, the Ideal Gas Law and the First Law of Thermodynamics (Rozier and Viennot, 1991; Yeo and Zadnik, 2001; Jasiem and Oberem, 2002; Loverude et al., 2002; Meltzer, 2004; Kautz et al., 2005a; Kautz et al., 2005b). The First Law of Thermodynamics connects energy transfers—that is, energy entering or leaving the system of interest—to changes in the internal energy of that system during some thermodynamic process (e.g., the compression of a gas in a cylinder kept at fixed temperature). The internal energy of the system is a property of the equilibrium state of that system, i.e., it depends on quantities such as the temperature, pressure, and volume of the system. Internal energy, and other quantities with similar features, are known as state functions. The energy transfer quantities in the First Law, work (mechanically transferred energy) and heat (thermally transferred energy), on the other hand, are not state functions; they only exist when a system is undergoing some sort of process. The important point here is that changes in state functions are independent of the thermodynamic process—one only needs to know the initial and final states to determine the change in internal energy of the system, for example. In contrast, the amount of work done on (or by) the system does depend on the process. Previous results regarding student understanding of the First Law have documented several common conceptual difficulties, such as indiscriminate application of the concept of a state function to non-state function quantities such as work and heat (Loverude et al., 2002; Meltzer 2004). Student Understanding of Physics and Calculus Concepts 3 For several years, members of the Physics Education Research Laboratory at the University of Maine (UMaine) have been exploring student learning in upper-level thermal physics courses, primarily taken by physics majors, extending the tools and results of research in introductory physics to the more specialized upper division. There exists a limited body of research in this area (Loverude et al., 2002; Meltzer, 2004; Meltzer, 2005b; Thompson et al., 2006; Cochran and Heron, 2006; Bucy et al., 2007; Mountcastle et al., 2007; Pollock et al., 2007; Christensen et al., 2009; Loverude 2009; Smith et al., 2009). Meltzer (2004) has suggested that particular difficulties experienced at the introductory level are also evident at the advanced undergraduate level. In thermal physics, as with most physics areas, specific mathematical concepts are required for a complete understanding and appreciation of the physics. Our main research agenda for the learning and teaching of thermal physics deals with identifying and addressing student conceptual difficulties with the physics content. A sub-theme of this research, highlighted here, investigates the extent to which any mathematical conceptual difficulties may affect students’ understanding of associated physics concepts in thermodynamics. In this area, as with many physics areas, specific mathematical concepts are required for a complete understanding and appreciation of the physics. Our research question in this context is: To what extent do students recognize, and/or understand, the relationship between the physics concepts and the underlying mathematics? Mathematics is a vital part of solving many physics problems, often used to condense a complicated conceptual problem into a relatively simple relationship between variables. Many convenient representational tools exist in physics (e.g., equations, graphs and diagrams) that simplify analysis of a complex problem. Appropriate interpretation of these representations Student Understanding of Physics and Calculus Concepts 4 requires recognition of the connections between the physics and the mathematics built into the representation and subsequent application of the related mathematical concepts (Redish, 2005). Meltzer (2002) has shown a link between mathematical acumen and success in an algebra-based physics course. Only a handful of studies in PER have attempted to investigate mathematical difficulties with calculus concepts among physics students (Thompson et al., 2006; Black and Wittmann, 2007; Bucy et al., 2007; Pollock et al., 2007; Rebello et al., 2007.) In thermodynamics, twoor three-dimensional graphical representations of physical processes are especially useful in helping to understand these processes. These diagrams can contain information about the thermodynamic “path” followed in a process, regions of different phase, and critical behavior. Graphs of pressure versus volume, known as P-V diagrams, are used extensively as representations of physical processes as well as of the corresponding mathematical models. Information can quickly and easily be obtained from these representations. For example, the work done on a system undergoing a thermodynamic process is defined as the integral of pressure with respect to the volume: